An electron beam can be split because the electron has an
additional property known as spin. The difference in the force on the electron
and the direction of spin is used to pull a beam with a mixture of electrons of
different spins apart. A simplified diagram of the apparatus is shown above. An
electron beam entering the apparatus is pulled apart and one spin orientation
is pulled up while the other is pulled down. An absorber placed so as to block
the path of the down going electrons is depicted above. With the absorber in
place the apparatus can be used to select electrons with a specific spin
orientation.

We will consider a similar apparatus but for photons.

Interpret as the probability. This will be used to make predictions
when we set up N identical experiments and predict the percentage of incident
particles that are observed after the beam passes through the filter.

We can of course choose any set of axises. Therefore

where , are just rotated by an angle wrt ,

Finally let us ask what happens if we were to change the
phase of the , waves by 90o ? The electric field rotates about
the z-axis.We can introduce this as an
alternative formulation in the following way.

We would like therefore to consider three bases which are
available to describe a photon propagating in the z-direction.

linearly polarized

,

(choose

linear polarized

rotated by

circularly polarized

To simplify the notation we will use the following

Photon is always traveling along the z-direction.

Choose 60 degree angle

Circularly polarization

The vector nature of the photon field is revealing its spin
structure. Photons are spin 1 particles. Indeed the E&M field carries
angular momentum and the basic quantum of the field carries it as well.
Interestingly, the vector structure of the E&M field is now a consequence
of the intrinsic spin structure of the photon. Later when we discuss spin we
will see that spin is related to an internal structure that requires a new
basis just as we have defined here in terms of polarization. We will also
discover that the general spin 1 particle can have three states of polarization
but the free photon because it is masseless has only two. Finally one can look
at a general field, for example the static Coulomb field around a charge. For
general fields the photon is not massless. It is referred to as a virtual
photon. It has the possibility of carrying all three states of polarization.
The electric field for example can have components along the propagation
direction (longitudinally polarized). Quantum development of the photon is
usually done within a relativistic framework. So we will go no further at this
point except to use the two-state nature of the photon to discuss quantum
behavior.

The way we shall proceed is by imagining projectors and
analyzers that are made up of the filters described above.For light we actually have simple filters
that select polarization.Any reasonable
optics lab will have beam splitters that can split a laser beam into any chosen
basis.

A general state that is some linear combination of will be transmitted
undisturbed through the open system.

A specific state |A> or |B> will projected onto the
output from any input. Put ina|A>
for a blocked –B filter and the amount a will emerge. Any |B> component will
be blocked.

This filter can be rotated so that the axes that determine
the meaning ofcan be changed from to with new basis states

Consider a system that has two quantum states.

For each of the above bases an apparatus can be built so
that any state will be split into its components and then recombined.The arrows indicate that the blocking
elements can either be inserted or removed.A and B represent the above states x,y;X,Y; or R,L

Such filter can be used as a filter that selects all or part
of the incident beam PROJECTOR or as an ANALYZER by measuring the transmission
for a given state either A or B. (Measure intensity with one of the paths
blocked).

We place three of these systems in a row:

The first filter will prepare the state. You can assume that
beam prepared by the first state is normalized to 100%.The second filter will select components and
the third filter will measure.How much
of the beam will be transmitted for the following situations.

Prepared state F 1

Selected state

Measured state

Result

type

blocking

x-y

none

x-y

none

x-y

Block x

x-y

Block x

X-Y

none

R-L

none

R-L

Block R

R-L

Block R

X-Y

Block Y

X-Y

Block Y

There are no surprises if all the filters are set as
described by case 1 to select a specific state, for example, .All the particle are
transmitted once the state I prepared by the first filter. So 100 % of the
prepared beam is measured.

Prepared state F 1

Selected state

Measured state

Result

type

blocking

1

x-y

none

100%

Measuring the orthogonal state again we are probably not
surprised will have a 0% of the beam measured.These results are the same no matter what analyzer is used as filter 2.

Rule 1

Open channels in the analyzer transmit the incident beam
unchanged.

Using a different basis for filter 3 provides results that
are based on the amplitudes squared.

Result of using the on the projected
states

Count photons emerging from the final filter.

The
above amplitudes build up as a result of the ensemble. We can reduce the
intensity so that one photon is in the system at a time. (Because photons
are bosons we can simultaneously put millions of photons through the
system at once. Thus we can simply measure the intensity of a transmitted
laser beam.)

We see
1 or 0 photons for any measurement (low intensity result). Photon must
interfere with itself.

Now let us examine the blocked analyzer.

Amplitude Squared

TOTAL

TOTAL

TOTAL

=1

Amplitude
for any path is a product of amplitudes at each step.

The diagram above illustrates that
there are many possible paths through the systems. Along each path I can define
amplitudes for each sector of the path. We start with an amplitude to arrive at
A1è
amp(enter at A1). Then we find the amplitudes to proceed from A1 to B1 è
amp(A1èB1).The amplitude for any path is the product of
amplitudes.

Path 1:

AMP1= amp(A1) amp(A1èB2)
amp(B2èC1)
amp(C1èD1)
amp(D1èX)

Path 2:

AMP2= amp(A1) amp(A1èB3)
amp(B3èC2)
amp(C2èD3)
amp(D3èX)

The
amplitude for a measured state is the sum of all amplitudes that reach
this state.

The amplitude for measuring a particle
at X is AMPX=sum over all the ways to reach X

The system above can be thought of as having a set of states
that span the space and are labeled as indicted in the drawing above.

èstate where the
particle goes through opening A1 and proceeds to all possible points.

èstate where the
particle goes through opening A2 and proceeds to all possible points.

èstate where the
particle goes through opening B2 from any conceivable starting point and
proceeds to all possible points.

All the A’s span the space since the particle must go
through an A opening. All the B’s and C’s span the space for the same reason.

would be any state
that goes through A1 and B2.

would be the amplitude
to go A1èB2èC1èD3èX.

If the states are truly complete then

Thus particles that impinge on walls must not be considered.
But within this framework any state can be represented by any basis and a
particle that goes through any one of the three available paths through a wall
can be described by an amplitude for each of these paths.

All of the remaining amplitudes can be determined by these
types of evaluation.

1

0

0

1

1

0

0

1

1

0

0

1

More details about light-----------------------------

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/emwavecon.html#c1

So to describe a photon in QM we would expect to introduce a
wavefunction that provides us with the location of the photons and a traveling
wave solution similar to the particle description. Light does have some special
properties due to the fact that it is massless it must travel at a constant
velocity c. But we can think the photon, for example, as having a momentum
state or position state just as a particle.

How does the photon relate to the observed macroscopic
field.In QM we know that the
observations are performed by looking at ensembles in order to extract what can
be known about the wavefunction. For Fermions nature requires us to repeat the
experiment multiple times because there is no possibility [Pauli exclusion
principle] to design an experiment with simultaneous multiple Fermions in the
same state. However for Boson a field can be built that has n identical
photons.A state prepared in this way
exhibits the ensemble average simultaneously. Consider a laser beam. It is, in
an ideal sense, a set of photons [billions] all in the same state. If you send
it through slits you see part of the beam pass through each slit. For quantum
mechanics we need to imagine that each individual photon must go through both
slits but when we view the experiment we instantaneously see the ensemble
average by placing a white card in the path of the beam or observe immediately
the interference pattern on a screen. Although the underlying rules for photon
and electron are the same the Fermion/Boson nature of these particles has
provided a very different view of the two phenomena matter/E&M. One that
until the advent of QM was assumed to be of a different character.

So now we are going to require that somehow each individual
photon carry the basic structure of the fields. Each photon, in some sense,
must be a traveling wave with both andfields present or alternatively a field with 4 components
through the potential formalism.

Let us go back to the basic structure of the traveling wave
and consider that it must be labeled by the electric field vector. If we pick a
direction of propagation then can be described by two independent possibilities, for
example, . What does this mean that it carries the vector nature of
the field? Somehow the basic building block, the photon, has some kind of
internal structure that provides an overall direction. This structure is two
dimensional in nature. We see that there must be two independent types of
photons an x and a y photon.Now any
general field can be described as a linear combination of the these two
photons.Let us turn back to classical
physics to discuss this feature of light.

Let us assume a traveling sine wave solution with an
Electric field pointing in the x direction. There is then an independent field
solution with its vector pointing in the y direction. A general direction for
the field can be reached simply by adding these two solutions in the same way
that usual vector components are added. The way that the E field depends on
position via the sinusoidal z dependence is preserved in the sum. A wave with
an electric field oscillating with a x-direction as a function of z plus an
equal wave oscillating up and down in the y direction as a sinusoidal function
of z give a vector pointing at 45o to the x-direction and
oscillating back and forth as as function of z.

So we will characterize the photon as having an “internal”
polarization. We fix the direction of propagation and then have an independent
2-state degree of freedom that allows us to define two different types of
photons, x-polarized and y-polarized.Should we want to describe a new photon with a polarization of a
different direction we simply combine the two states.

First we consider what is a photon?One beneficial aspect of QM is that it
unifies its treatment of particles and fields. In classical physics one has a
description of matter and a separate description of the Electric and Magnetic
field. In QM the same formalism is used to describe both phenomena. Matter is
different than fields because of the properties possessed by the different
entities.One finds different types of
matter also.For example, there are
quarks and leptons that comprise the known forms of matter. The quarks and
leptons are further divided by properties such as mass, charge, and color.So the photon and the electron are two
elements of quantum theory but with different characteristics but which are
confined by the same behavior. Let us look at the critical differences:

property

electron

photon

mass

small (0.5 MeV=mass
energy)

0 (free photons)

spin

1/2

1

statistics

Fermion

Boson

Pauli property

one per state

many per state

The type of photon we will be discussing is the free photon.
In classical physics Maxwell’s equation can describe a field in a region where
there are no charges or currents.

These equations
are normally reworked to form a wave equation for the electric and magnetic
field.

Light is then understood as a traveling Electric and
magnetic disturbance. In particular it is interesting to look for the special
solutions to the wave equation that exhibit sinusoidal behavior.This is a basis to describe all traveling
waves in terms of a mixture of light of various colors (frequency).

The magnetic field accompanies the electric field. The
magnetic field is perpendicular. The relative position of the magnetic field
will determine the direction of orientation. Both and are perpendicular to the direction of propagation. So for
every there is a known . One can then characterize a traveling light wave by its
frequency, direction and Electric field vector.

One might also remember that the E&M can be described
using two potentials. is the vector potential which is used to obtain . And V is a scalar potential that is introduced to describe .

The traditional quantum formulation is to build a 4-vector
from these two fields